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%I #46 May 17 2020 11:16:20
%S 2,16,7936,11184128,209865342976,2475749026562048,
%T 123460740095103991808,5779796046952399460368384,
%U 3729407703720529571097509625856,485491405392529556189699853976076288,193817991886041515914007312001087567822848,56920344782482721622150071084079041150980194304
%N a(n) = numerator (2^(4*n-1) * (2^(4*n-2) - 1) * (Bernoulli(4*n-2) / (4*n-2)!) * ((2*n-2)! / Euler(2*n-2))^2).
%C For every s odd power of odd prime number p applies: pi(p; 4, 3) = pi(p^s; 4, 3) and pi(p; 4, 1) = pi(p^s; 4, 1).
%C Product_{p = A002144} ((p^(2*n - 1) - 1) / (p^(2*n - 1) + 1)) = (2^(2*n) + 2) * (2*n - 2)! * (Pi^(2*n - 1) / zeta(2*n - 1)) * (zeta(4*n - 2) / Pi^(4*n - 2)) / abs(EulerE(2*n - 2)), n > 1.
%C Product_{p = A002145} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1)) = (2^(2*n) - 2) * (2*n - 2)! * (zeta(2*n - 1) / Pi^(2*n - 1)) / abs(EulerE(2*n - 2)), n > 1.
%C Product_{p = A065091, m_p = (p mod 4) - 2} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1))^m_p) = (2^(4*n - 1) * (2^(4*n - 2) - 1) * (BernoulliB(4*n - 2) / (4*n - 2)!) * ((2*n - 2)! / EulerE(2*n - 2))^2 ) = a(n) / A334835(n).
%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html">Some Constants from Number theory</a>
%F a(n) = numerator (Product_{p = A065091, m_p = (p mod 4) - 2} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1))^m_p) = numerator (2^(4*n) - 4) * ((2*n - 2)! / EulerE(2*n - 2))^2 * (zeta(4*n - 2) / Pi^(4*n - 2))).
%F From _Vaclav Kotesovec_, May 17 2020: (Start)
%F a(n) / A334835(n) ~ 1 as n tends to infinity.
%F a(n) = numerator((1 - 1/2^(4*n-2)) * zeta(4*n-2) / DirichletBeta(2*n-1)^2). (End)
%t Numerator[Table[2^(4*s - 1) * (2^(4*s - 2) - 1) * BernoulliB[4*s - 2] * (2*s - 2)!^2 / (EulerE[2*s - 2]^2 * (4*s - 2)!), {s, 1, 15}]] (* or *) Numerator[Table[(1 - 1/2^(4*s - 2))*Zeta[4*s - 2]/DirichletBeta[2*s - 1]^2, {s, 1, 15}]] (* _Vaclav Kotesovec_, May 17 2020 *)
%o (PARI) E(n) = subst(bernpol(2*n+1), 'x, 1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1); \\ see A000364
%o a(n) = numerator((2^(4*n-1)*(2^(4*n-2)-1)*(bernfrac(4*n-2)/(4*n-2)!)*((2*n-2)!/ E(n-1))^2)); \\ _Michel Marcus_, May 17 2020
%Y Cf. A000040, A065091, A002144, A002145, A334835 (denominators).
%Y Cf. A000364, A027641/A027642.
%K nonn,frac
%O 1,1
%A _Dimitris Valianatos_, May 16 2020
%E More terms from _Michel Marcus_, May 17 2020
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